finite set

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finite set
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In mathematics, a finite set is a set that has a (wikt:finite|finite) number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example,
is a finite set with five elements. The number of elements of a finite set is a natural number (a non-negative integer) and is called the cardinality of the set. A set that is not finite is called infinite. For example, the set of all positive integers is infinite:
Finite sets are particularly important in combinatorics, the mathematical study of counting. Many arguments involving finite sets rely on the pigeonhole principle, which states that there cannot exist an injective function from a larger finite set to a smaller finite set.

Definition and terminology

Formally, a set {{mvar|S}} is called finite if there exists a bijection
fcolon Srightarrow{1,ldots,n}
for some natural number {{mvar|n}}. The number {{mvar|n}} is the set's cardinality, denoted as |{{mvar|S}}|. The empty set {} or Ø is considered finite, with cardinality zero.{{harvtxt|Apostol|1974|p=38}}{{harvtxt|Cohn|1981|p=7}}{{harvtxt|Labarre|1968|p=41}}{{harvtxt|Rudin|1976|p=25}}If a set is finite, its elements may be written — in many ways — in a sequence:
x_1,x_2,ldots,x_n quad (x_i in S, 1 le i le n).
In combinatorics, a finite set with {{mvar|n}} elements is sometimes called an {{mvar|n}}-set and a subset with {{mvar|k}} elements is called a {{mvar|k}}-subset. For example, the set {5,6,7} is a 3-set â€“ a finite set with three elements â€“ and {6,7} is a 2-subset of it.(Those familiar with the definition of the natural numbers themselves as conventional in set theory, the so-called von Neumann construction, may prefer to use the existence of the bijection fcolon Srightarrow n, which is equivalent.)

Basic properties

Any proper subset of a finite set S is finite and has fewer elements than S itself. As a consequence, there cannot exist a bijection between a finite set S and a proper subset of S. Any set with this property is called Dedekind-finite. Using the standard ZFC axioms for set theory, every Dedekind-finite set is also finite, but this implication cannot be proved in ZF (Zermelo–Fraenkel axioms without the axiom of choice) alone. The axiom of countable choice, a weak version of the axiom of choice, is sufficient to prove this equivalence.Any injective function between two finite sets of the same cardinality is also a surjective function (a surjection). Similarly, any surjection between two finite sets of the same cardinality is also an injection.The union of two finite sets is finite, with
|Scup T| le |S| + |T|.
In fact:
|Scup T| = |S| + |T| - |Scap T|.
More generally, the union of any finite number of finite sets is finite. The Cartesian product of finite sets is also finite, with:
|Stimes T| = |S|times|T|.
Similarly, the Cartesian product of finitely many finite sets is finite. A finite set with n elements has 2{{sup|n}} distinct subsets. That is, thepower set of a finite set is finite, with cardinality 2{{sup|n}}.Any subset of a finite set is finite. The set of values of a function when applied to elements of a finite set is finite.All finite sets are countable, but not all countable sets are finite. (Some authors, however, use "countable" to mean "countably infinite", so do not consider finite sets to be countable.)The free semilattice over a finite set is the set of its non-empty subsets, with the join operation being given by set union.

Necessary and sufficient conditions for finiteness

In Zermelo–Fraenkel set theory without the axiom of choice (ZF), the following conditions are all equivalent:{{Citation needed|date=October 2009}}
  1. S is a finite set. That is, S can be placed into a one-to-one correspondence with the set of those natural numbers less than some specific natural number.
  2. (Kazimierz Kuratowski) S has all properties which can be proved by mathematical induction beginning with the empty set and adding one new element at a time. (See below for the set-theoretical formulation of Kuratowski finiteness.)
  3. (Paul Stäckel) S can be given a total ordering which is well-ordered both forwards and backwards. That is, every non-empty subset of S has both a least and a greatest element in the subset.
  4. Every one-to-one function from P(P(S)) into itself is onto. That is, the powerset of the powerset of S is Dedekind-finite (see below).The equivalence of the standard numerical definition of finite sets to the Dedekind-finiteness of the power set of the power set was shown in 1912 by {{harvnb|Whitehead|Russell|2009|p=288}}. This Whitehead/Russell theorem is described in more modern language by {{harvnb|Tarski|1924|pp=73–74}}.
  5. Every surjective function from P(P(S)) onto itself is one-to-one.
  6. (Alfred Tarski) Every non-empty family of subsets of S has a minimal element with respect to inclusion.{{harvnb|Tarski|1924|pp=48–58}}, demonstrated that his definition (which is also known as I-finite) is equivalent to Kuratowski's set-theoretical definition, which he then noted is equivalent to the standard numerical definition via the proof by {{harvnb|Kuratowski|1920|pp=130–131}}. (Equivalently, every non-empty family of subsets of S has a maximal element with respect to inclusion.)
  7. S can be well-ordered and any two well-orderings on it are order isomorphic. In other words, the well-orderings on S have exactly one order type.
If the axiom of choice is also assumed (the axiom of countable choice is sufficientBOOK,weblink Handbook of Differential Equations: Ordinary Differential Equations, Canada, A., Drabek, P., Fonda, A., 2005-09-02, Elsevier, 9780080461083, en, {{Citation needed|date=September 2009}}), then the following conditions are all equivalent:
  1. S is a finite set.
  2. (Richard Dedekind) Every one-to-one function from S into itself is onto.
  3. Every surjective function from S onto itself is one-to-one.
  4. S is empty or every partial ordering of S contains a maximal element.

Foundational issues

Georg Cantor initiated his theory of sets in order to provide a mathematical treatment of infinite sets. Thus the distinction between the finite and the infinite lies at the core of set theory. Certain foundationalists, the strict finitists, reject the existence of infinite sets and thus recommend a mathematics based solely on finite sets. Mainstream mathematicians consider strict finitism too confining, but acknowledge its relative consistency: the universe of hereditarily finite sets constitutes a model of Zermelo–Fraenkel set theory with the axiom of infinity replaced by its negation.Even for those mathematicians who embrace infinite sets, in certain important contexts, the formal distinction between the finite and the infinite can remain a delicate matter. The difficulty stems from Gödel's incompleteness theorems. One can interpret the theory of hereditarily finite sets within Peano arithmetic (and certainly also vice versa), so the incompleteness of the theory of Peano arithmetic implies that of the theory of hereditarily finite sets. In particular, there exists a plethora of so-called non-standard models of both theories. A seeming paradox is that there are non-standard models of the theory of hereditarily finite sets which contain infinite sets, but these infinite sets look finite from within the model. (This can happen when the model lacks the sets or functions necessary to witness the infinitude of these sets.) On account of the incompleteness theorems, no first-order predicate, nor even any recursive scheme of first-order predicates, can characterize the standard part of all such models. So, at least from the point of view of first-order logic, one can only hope to describe finiteness approximately.More generally, informal notions like set, and particularly finite set, may receive interpretations across a range of formal systems varying in their axiomatics and logical apparatus. The best known axiomatic set theories include Zermelo-Fraenkel set theory (ZF), Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC), Von Neumann–Bernays–Gödel set theory (NBG), Non-well-founded set theory, Bertrand Russell's Type theory and all the theories of their various models. One may also choose among classical first-order logic, various higher-order logics and intuitionistic logic.A formalist might see the meaning{{fact|date=April 2017}} of set varying from system to system. Some kinds of Platonists might view particular formal systems as approximating an underlying reality.

Set-theoretic definitions of finiteness

In contexts where the notion of natural number sits logically prior to any notion of set, one can define a set S as finite if S admits a bijection to some set of natural numbers of the form {x , | , x

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